1. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 1 DISCRETE SIGNALS AND SYSTEMS Sequences Exponential Sequences s is a complex number If s is a real number and a=e then the sequence is called real exponential

2. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 2 DISCRETE SIGNALS AND SYSTEMS Sequences Geometric Sequence: real exponential sequence defined as:

3. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 3 DISCRETE SIGNALS AND SYSTEMS Sequences Sinusoidal sequence

4. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 4 DISCRETE SIGNALS AND SYSTEMS Properties of Sequences Sum of two signals: w = x + y w(k) = x(k) + y(k) Multiplication of two signals: w = x. y w(k) = x(k) y(k) Multiplication of signal by a scalar: w = c x w(k) = c x(k) Energy of signal: If a signal is delayed by m time units then x(k) becomes x(k-m) Sequence: sum of scaled, delayed unit samples

5. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 5 DISCRETE SIGNALS AND SYSTEMS Properties of Sequences Example:

6. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 6 DISCRETE SIGNALS AND SYSTEMS Signal Measures The signal norm is defined: Some properties of the signal norm are: Norm 1: Sum of the magnitudes of each signal sample It is used to determine system stability

7. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 7 DISCRETE SIGNALS AND SYSTEMS Signal Measures Norm 1: Sum of the magnitudes of each signal sample. It is used to determine system stability. Norm 2: Provides a measure of the signal power. It is the most frequent used measure. Norm infinity: Gives the peak magnitude of the signal.

8. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 8 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Discrete–time system: Converting input sequence x=x(n) into output sequence y=y(n) through transformation φ[.] y(n) = φ[x(n)] Α linear system is defined by the principle of superposition. If, Then a system is linear if and only if,

9. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 9 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Example 1: Is the following system linear? y(n) = 10x(n) - 5y(n-1) ___________________________________________ ____________________________________________ Yes, the system is linear

10. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 10 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Example 2: Is the following system linear? _________________ ______________________________________ No, the system is not linear

11. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 11 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems A system is time–invariant or shift- invariant if, y(n) is response to x(n) then y(n-k) is response to x(n-k) : a signal delay of k samples Example 1: Is the following system shift-invariant? y(n) = 10x(n) - 5y(n-1) ________________________ _________________________ Yes, the system is shift-invariant

12. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 12 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems Example 2: Is the following system shift-invariant? y(n) = n x(n) ______________ __________ No, the system is not shift-invariant We said that we can express: The system output response is:

13. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 13 DISCRETE SIGNALS AND SYSTEMS Linear, Shift-Invariant Systems If the system is linear, the response of the system to a sum of inputs is the same as the sum of the system’s responses to each of the individual inputs: By definition: φ[δ(k)] = h(k) If the system is shift-invariant: φ[δ(n-k)] = h(n-k)  If a system is linear and shift-invariant, the convolution sum applies y(n) = x(n) * h(n)

14. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 14 DISCRETE SIGNALS AND SYSTEMS Linear Convolution The graph method of computing the Convolution sum  Folding one of the sequences x(n) or h(n) over the horizontal axis and getting x(-k) or h(-k)  Shifting the folded sequence creating x(n-k) or h(n-k)  The addition of the product of the two sequences at time n yields the output y(n) Example: What is the response y(n) if h(n)={1,2,3} and x(n) = {3,1,2,1}

15. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 15 DISCRETE SIGNALS AND SYSTEMS Linear Convolution As a result, y(n)={3,7,13,8,8,3}  If x(n): N samples, h(n): M samples, y(n): N+M-1 samples

16. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 16 DISCRETE SIGNALS AND SYSTEMS Stability and Causality A system is stable if a bounded input produces a bounded output. Necessary and sufficient condition This is the norm as defined in a previous session Example: is the following system stable: y(n) = x(n) + b y (n-1) Calculating the norm we have: The system is stable

17. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 17 DISCRETE SIGNALS AND SYSTEMS Stability and Causality A causal system is a system that at time m produces a system output that is depended only on current and past inputs, that is: n<m This is always true for a unit impulse response it is zero for n<0  A discrete-time, linear, shift-invariant system is causal if and only if h(n)=0 for n<0 Example: is the following system causal: y(n) = x(n) + b y (n-1) Since the unit-sample response is zero for n<0, the system is causal.

18. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 18 DISCRETE SIGNALS AND SYSTEMS Digital Filters A broad class of digital filters are discribed by linear, constant coefficient, difference equations {a i } {b i } characterize the system Given: initial conditions x(i), y(i) i=-1,-2,…,-M input sequence: x(n) output sequence: y(n) The system is causal. The system is Mth-order. Two main classes of digital filters:  Infinite Impulse Response (IIR)  Finite Impulse Response (FIR)

19. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 19 DISCRETE SIGNALS AND SYSTEMS Digital Filters  Infinite Impulse Response (IIR): current and past input samples and past output samples. Example: Determine impulse response for the first- order IIR filter. y(n) = x(n) + b y (n-1) Assume: x(n)=0, y(n)=0 for n<0 x(n)=δ(n) h(n)=δ(n) + b h(n-1) h(n)=0 n<0 h(0)=1 + b 0 = 1 h(1)=0 + b 1 = b h(2)=0 + b b = b 2 h(3)=0 + b b 2 = b 3 …….. h(n)= b n u(n)

20. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 20 DISCRETE SIGNALS AND SYSTEMS Digital Filters  Finite Impulse Response (FIR): current and past input samples.  The coefficients of the FIR filter are equivalent to the filter’s impulse response. Why? Remember convolution? h(k) = b k k=0,1,2,…,M h(k) = 0 otherwise